The Exciting Universe Of Music Theory

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Scale 105

Scale 105, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).



Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11


Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 705


A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)


An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.



Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.


Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

prime: 75


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[3, 2, 1, 6]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<1, 1, 2, 0, 1, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.


Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,3,6}
<2> = {3,5,7,9}
<3> = {6,9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.


Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.


Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.


Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.



Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".


Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(4, 3, 18)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Diminished Triads{0,3,6}000

The following pitch classes are not present in any of the common triads: {5}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.


Modes are the rotational transformation of this scale. Scale 105 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 525
Scale 525, Ian Ring Music Theory
3rd mode:
Scale 1155
Scale 1155, Ian Ring Music Theory
4th mode:
Scale 2625
Scale 2625, Ian Ring Music Theory


The prime form of this scale is Scale 75

Scale 75Scale 75: Iloian, Ian Ring Music TheoryIloian


The tetratonic modal family [105, 525, 1155, 2625] (Forte: 4-13) is the complement of the octatonic modal family [735, 1785, 1995, 2415, 3045, 3255, 3675, 3885] (Forte: 8-13)


The inverse of a scale is a reflection using the root as its axis. The inverse of 105 is 705

Scale 705Scale 705: Edrian, Ian Ring Music TheoryEdrian


Only scales that are chiral will have an enantiomorph. Scale 105 is chiral, and its enantiomorph is scale 705

Scale 705Scale 705: Edrian, Ian Ring Music TheoryEdrian


In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 105       T0I <11,0> 705
T1 <1,1> 210      T1I <11,1> 1410
T2 <1,2> 420      T2I <11,2> 2820
T3 <1,3> 840      T3I <11,3> 1545
T4 <1,4> 1680      T4I <11,4> 3090
T5 <1,5> 3360      T5I <11,5> 2085
T6 <1,6> 2625      T6I <11,6> 75
T7 <1,7> 1155      T7I <11,7> 150
T8 <1,8> 2310      T8I <11,8> 300
T9 <1,9> 525      T9I <11,9> 600
T10 <1,10> 1050      T10I <11,10> 1200
T11 <1,11> 2100      T11I <11,11> 2400
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 75      T0MI <7,0> 2625
T1M <5,1> 150      T1MI <7,1> 1155
T2M <5,2> 300      T2MI <7,2> 2310
T3M <5,3> 600      T3MI <7,3> 525
T4M <5,4> 1200      T4MI <7,4> 1050
T5M <5,5> 2400      T5MI <7,5> 2100
T6M <5,6> 705      T6MI <7,6> 105
T7M <5,7> 1410      T7MI <7,7> 210
T8M <5,8> 2820      T8MI <7,8> 420
T9M <5,9> 1545      T9MI <7,9> 840
T10M <5,10> 3090      T10MI <7,10> 1680
T11M <5,11> 2085      T11MI <7,11> 3360

The transformations that map this set to itself are: T0, T6MI

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 107Scale 107: Ansian, Ian Ring Music TheoryAnsian
Scale 109Scale 109: Amsian, Ian Ring Music TheoryAmsian
Scale 97Scale 97: Athian, Ian Ring Music TheoryAthian
Scale 101Scale 101: Apoian, Ian Ring Music TheoryApoian
Scale 113Scale 113, Ian Ring Music Theory
Scale 121Scale 121: Asoian, Ian Ring Music TheoryAsoian
Scale 73Scale 73: Diminished Triad, Ian Ring Music TheoryDiminished Triad
Scale 89Scale 89: Aggian, Ian Ring Music TheoryAggian
Scale 41Scale 41: Vietnamese Tritonic, Ian Ring Music TheoryVietnamese Tritonic
Scale 169Scale 169: Vietnamese Tetratonic, Ian Ring Music TheoryVietnamese Tetratonic
Scale 233Scale 233: Bigian, Ian Ring Music TheoryBigian
Scale 361Scale 361: Bocritonic, Ian Ring Music TheoryBocritonic
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 1129Scale 1129: Raga Jayakauns, Ian Ring Music TheoryRaga Jayakauns
Scale 2153Scale 2153: Navian, Ian Ring Music TheoryNavian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler ( used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.